How to insert a PNG as a plane in the scene, but make it so it's ALWAYS in front (and never clips into the other objects)?
Okay, I think I know something that will work (It seems so obvious now, but I'll leave the question up in case someone else has the same problem).

I'll put the PNG's in the scene and just deal with the clipping while I'm animating, then render them separately from the background, and composite them on top afterwards. I honestly kind of feel silly for not thinking of it sooner.